Stability of laminar monotone shear flows in a channel for high Reynolds number

TL;DR

This paper establishes conditions under which a laminar shear flow in a channel remains stable at high Reynolds numbers, generalizing previous results by including long wave perturbations and allowing certain flow profiles.

Contribution

It provides a new stability criterion for monotone shear flows allowing vanishing second derivatives, extending prior work to include long wave perturbations.

Findings

Flow stability is guaranteed if a specific operator is strictly positive for all relevant parameters.

The results apply to flows with vanishing second derivatives, broadening the class of stable flows.

Long wave perturbations are incorporated into the stability analysis.

Abstract

We consider the stability of a laminar flow $U\in C^4([-1,1])$ in the two-dimensional channel $\mathbb{R} \times[-1,1]$ in the large Reynolds number limit. Assuming that $U$ is strictly monotone but allowing $U^{\prime\prime}$ to vanish, we obtain that if the operator $$ {\mathcal K}_{\nu}=-\frac{d^2}{dx^2}+\frac{U^{\prime\prime}}{U-\nu} \,, $$ is strictly positive for all $\nu\in\mathbb{R}$ for which $U^{\prime\prime}(U^{-1}(\nu))=0$,then $U$ is stable for sufficiently large Reynolds number. This contribution generalizes previous results mostly by allowing long wave perturbations (but much shorter than the Reynolds number).